The potential for disruption and the importance of a pricing mechanism which can shine light on healthcare prices, home to the most chaotic price-setting formula across any consumer landscape, cannot be emphasized enough. In the current US healthcare market, consumers operate entirely in the dark. They have no knowledge (with or without insurance) of what they will pay for goods & services before they buy those goods & services; even the most sophisticated consumers face an inconceivably convoluted black box. Of course, this is by design—the healthcare system in this country is incentive mis-aligned. This is true of the most basic care products (e.g. vaccines, prescription drugs, childbirth), elective procedures (e.g. cortisone epidurals, braces, laparoscopic hernia surgeries), and clinical necessities (e.g. oncology, dialysis, burn treatment). This has led to a constant increase in care provider (from the physician-level to the hospital system-level) prices out-of-pocket, the highest in any consumer category across any country, as free market competitive dynamics play little to no role in price-setting—the demand curve is artificially skewed inelastic. Treatment pricing, et ceteris paribus, varies by orders of magnitude between locations in the same geography. On the other hand, this implies that clarity in this landscape lends itself to an unimaginably large arbitrage opportunity, to the tune of billions.

However, in addition to this natural consumer benefit, price transparency and the stabilization of the demand curve contributes to system level efficiency gains as well, via a contraction of dead weight loss. Though fair competitive dynamics will eat into healthcare producer surplus, given its quasi-oligopolistic nature, they also incentivize the optimal allocation of resources and attention throughout the supply chain. Otherwise stated, treatment prices will drop across the system level, but differentiation outside of the pricing mechanism between providers will become more apparent. In this type of transparent system, softer factors like care quality, medical ingenuity, or nursing professionalism (to name a few), become significant and “less noisy” signals, underwritten by price differences between providers for the same treatment (and plan). We have seen this type of efficient market transformation play out across a number of industries, including transportation, real estate, and food/dining. Undoubtedly, our healthcare system will always enjoy a certain degree of inelasticity compared to some of these other industries, since several medical treatments are viewed by consumers and/or professionals as life-threateningly necessary, but critically, most out-of-pocket spend annually is directed towards commodities and chronic care (volume-adjusted). A model like the one described in this paper, consequently, may very well serve as a cornerstone for the healthy, fair allocation of goods across consumers—a consumer market that actually functions like one.

The key to our development of a theoretical model which can accurately price current care treatments, within a probabilistic distribution of noise εαC, is to (1) map an accurate, general form of the price/volume distribution ex-ante and (2) determine the correct price to charge all consumers across the distribution, where the "correct" price is one that no consumer c who would have transacted at their original price would not transact on now. In addition, given that the true price, the price which is paid to the care provider, is varied, the modeled price must break-even in aggregate across the true price distribution. In the theoretical case, we assume that the true price distribution is identical to the distribution derived in (1). These conditions ought to sustain on a per-provider, per-treatment and system-level basis (all treatments across all providers). An important caveat to (2): an advantage to the design of such a model for US Healthcare specifically is that no consumer can practically ever predict or know their individual true price of treatment before receiving their bill. Thus, if the system were to assume financial responsibility on behalf of the consumer, the modeled price must really only satisfy an absurdity doctrine or "sniff test"—it must feel reasonable to pay. Concretely, the modeled price must exist within a band n standard deviations away from the E[V] of the treatment, with n denoting the upper bound of reason for the most price elastic consumer in C*, i.e. the max price anyone could conceivably be willing to pay given a treatment's weighted average price. In the Model section, we break down our approach to both of these conditions, discussing three different possible directions, each with their advantages and disadvantages, though all of them are sufficient standalone.

1. Across all three of the models explored, we use the same general form for the distribution, from here on denoted as f(x). To build f(x) for the current year, where x is our price axis and f(x) is our volume output, we choose to leverage historical claims data for any priceαC. Here, we can control for a myriad of confounding factors, primarily YoY inflation and federally mandated floors and ceilings. Though these principle confounding variables undoubtedly contribute to most of year over year distribution fluctuations, additional complexity is inevitable, which we discuss in detail in the following sub-section. To exhaustively satisfy condition (2), we solve open-form, for any f(x), regardless of what empirical claims data finally imputes, with practical and intuitive constraints. Namely, f(x) is a true function, f(x) ≥ 0 for all x, and f(x) is continuous (and thereby, differentiable). There exists a single treatment volume for any treatment price, and there are no negative treatment volumes. For intuition's sake, we can reference a normal curve subject to these constraints, the most likely conclusion to any claims data analysis—there is a mode within f(x) and the curve increases and decreases around the global maximum. It is interesting to note here that such a curve does not possess the typical, monotonically decreasing property that Keynesian economics suggests, illustrating the divergence from free market dynamics within US healthcare. As an aside: there is also a case to be made around the possibility that f(x) could be bi-modal for certain treatments, though any solution ought to extend to such a case naturally, given that it satisfies the same constraints.
2. The base case in arriving on a deterministic priceαC, from here denoted as z, is simply finding the weighted average price of the treatment within f(x). The system breaks-even at z by default, on aggregate, paying down per-treatment losses with per-treatment gains. On top of this, the system also accrues interest (value external to the system) on z × f(x) assuming the consumer transacts and there is some time period till their "true" bill is due. Though there is some marginal inter-state variation, the typical healthcare grace period for payment is 60-90 days, on top of the 15-60 days it may take for the provider system to process insurance & claims to construct the final bill for the consumer. Consequently, our first model derives a set price z, the weighted average price of f(x), less all value external to the system—assuming the system is not profiting from the interest. Of course, such a model is necessarily non-pareto optimal and subjects the system to the burden of potential customer dissatisfaction upon receipt of their "true" bill. Any consumer who ends up paying more (where z > xαC), will naturally feel slighted. Though there is a case to be made that the consumer could not have known where on the distribution f(x) they would have landed, it is challenging to argue this case on a per-consumer basis. Patently, this can be mitigated entirely by preventing the consumer from ever seeing their final bill in the first place, that is, assuming financial responsibility in the transaction. We explore such avenues for implementation, adjusting for this weakness, in the Model and Analysis sections.

Outside the base case, we also posit a more extreme, optimistic form for the model, designing for pareto-optimality. In this case, any consumer whose final bill is less than or equal to the modeled price (where xαC ≤ z), gets reimbursed for the difference, and any consumer whose final bill is greater than or equal to the modeled price (xαC > z) pays no higher than the modeled price—the system absorbs the cost and pays it. Here, the system's cost basis is far higher and the only value generated is external to the transaction, in the form of interest. Put simply, the interest on GMV (general merchandise volume, i.e. transaction volume * price per transaction) for the entire consumer population is used to pay out all consumers for whom xαC > z. Fortunately, a viable z necessarily exists, given that at a 0% interest rate, z = the max possible priceαC. The primary constraint of this model is its disproportionate dependence on the interest rate, limiting the extent to which z might reduce to "reasonable" levels. Still, for distributions with low variance, such a model is extremely well-suited.

Lastly, the final model we consider is a dynamic hybrid of the aforementioned models, where the reimbursement or rebate is scaled by a maneuverable θ ∈ [0, 1] coefficient of the full amount. Such an approach can incur the advantage of the first two models, while also mitigating downside risk from eroding consumer trust because of inaccurate charging. Implicitly, models one and two are subsets of this dynamic hybrid, respectively, where θ = 0 (no rebate) or θ = 1 (full rebate).

Though a theoretical solvent general model is a massive step in the right direction, there are practical limitations to its execution. Namely, additional YoY confounding to healthcare claims distribution, limited claims volume on certain treatments, and balance sheet risk from interest accrual. We discuss these complexities in detail and subsequently provide some analysis mitigating each of these concerns.

Additional YoY confounding in claims

Any historical predictions on claims are likely going to be inaccurate for a number of reasons. Controlling across a broad time horizon, we can eliminate some separable pieces, including cyclical market contractions and inflation, with relative ease. That said, we expect to see some additional deviation from factors such as renewed hospital-insurer negotiations, changes to the hospital chargemaster, consumer health shocks (e.g. Covid), medical innovation (newer therapeutics and hardware), and more. While it seems intuitively reasonable to expect that the magnitude of these confounding variables will be modest in scope, especially across a large population group, they can never be modeled entirely; to do so would be tantamount to predicting the “future of health”. However, the presence of such confounding necessarily decreases over time. Given that the system can dynamically adapt to new inputs and begin ingesting “true” claims real-time, we can weight down the historical heuristic simultaneously. We detail the requisite algorithmic means of accomplishing this, via back-propagation, in our Analysis section.

Limited claims volume on specific treatments

Similar to additional confounding on f(x) limiting its predictive accuracy, limited claims volume on some rare subset of treatments prevents accurate distribution mapping. Let us assume that there are less than a hundred heart surgeries that happen per year in most geographic subsets; given sufficiently high variance within pricing, mapping a continuous distribution proves challenging. As a result, stabilized pricing incurs a proportionally higher tail risk. Fortunately, there are 4 excellent mitigating factors: 1) Private/Federal out of pocket maximums limit blow-up risk for most expensive high-variance surgeries within the US consumer population (the system can always price at the max claim price to take even more advantage of this), 2) Given their low volume, high-variance surgeries contribute proportionally less to system-level GMV, 3) Back-propagation of new claims data exclusively improves system-level accuracy over time, and most importantly, 4) Treatments are extremely price-fungible across similar price domains and variances. Otherwise stated, we can group treatments with similar price distributions into sets—f(x)α per treatment can extend to any f(x)α* for a set of treatments. Such matching allows us to benefit from higher volumes across a treatment group even when individual treatments in that group occur rarely, optimizing system-level accuracy. Notably, this may result in treatments which are not adjacent (e.g. PET scan with contrast and Laparoscopic Hernia Surgery) being priced together. Given that the price domains overlap ex-ante, there is little reason to expect this will have an impact on the transaction experience or willingness to pay.

Balance sheet risk

Any system which generates interest from structural float takes on a variable amount of balance sheet risk. Naturally, the higher the requisite (or desired) yield, the more risk. However, such interest rate risk can be constructed as both a symmetric merit and weakness of all modeled systems—thus, portfolio risk management ought to be a priority, which we discuss in our Interest Rate Variance sub-section. In addition, private/federal out of pocket maximums (as above) limit the tail risk here as well. Finally, it is important here to note that the modeled systems in this paper are not “profit-seeking”, and thus the burden of yield generation becomes less relevant outside of its impact on consumer utility (i.e. increasing interest yield ought to decrease z, naively increasing their utility from transacting).

In this base case model, we find the lowest stable price z, assuming no consumer receives a rebate if their final price is above or below z. Thus, we find z where all GMV through the system priced at z where z > x, plus interest to the system of all GMV, is greater than or equal to the loss accrued where x > z:

For the model to not profit-seek i.e. balancing to break-even:

Given the nature of definite integration, this serves as a stable open-form solution, where ∂x/∂z = 0. In addition, this solution yields some directional relationships for z, namely, ∂i/∂z < 0, ∂l/∂z > 0. As expected, higher interest rates reduce z linearly and a higher max price increases z proportionally to f(x).

Intuitively, this simplifies discretely, to the weighted average price across f(x) less weighted average interest on the stable price across f(x):

Or,

With a full rebate, pareto-optimal, we find the the lowest stable price z where the interest on the GMV of the system alone must be greater than or equal to the loss accrued by the system, where x > z. Thus, the following relationship must hold:

For the model to not profit-seek i.e. balancing to break-even:

Or, integrating by parts, ź s.t.,

where F(x) is the antiderivative and f’(x) is the derivative of f(x). Once again, this serves as a stable open-form solution given the nature of definite integration. To simplify discretely, we sum over an increment of z; to simplify definitely, we can substitute in constants for i and l, and a specific function over a definite domain for f(x).

This final model, a superset of the two aforementioned, can build on both pricing models with the inclusion of a θ ∈ [0, 1]. When θ = 1, the system returns zero rebate (model 1), and when θ = 0, the system returns a full rebate (model 2):

Discretized Equation

Summation, where zn+1 = zn + 0.01 (by cent).

Example of a Definite Solution

Now, to arrive at a definite, integer solution from our continuous model, let us substitute sin(x) in our final dynamic model. As discussed, the dynamic model is necessarily inclusive of the first two models (no rebate and full rebate). We constrain the domain, x ∈ [0, π] to satisfy our desired conditions, and integrate by parts. Notably, sin(x) in this domain arrives at a reasonable approximation of a normal curve:

To simplify, we take the definite derivatives of both sides of the equation, where f(x) = sin(x), and substitute the constants; i = 10% and θ = 0 (full rebate, eliminating the first term).

Solving for z,

When ź is a real number,

Graphically,

As we can see, the definite solution, 1.81, lies only slightly past the midpoint π/2 of the domain, a reasonable price. Moreover, this stable price breaks-even across the distribution should all consumers transact, inclusive of all noise.

The Bootstrap Problem

Though the open-form model generates discrete/definite form solutions when implemented (as seen above), there is still a dangerous implicit assumption made within the system regarding time independence: all transactions occur along the distribution simultaneously so interest is generated over a constant, non-trivial time t before final payment is due. Clearly, any realistic implementation of such a model does not sustain such an assumption. Fortunately, we can accommodate such a bootstrap problem easily—with an a priori “liquidity pool”, before the float from care transaction GMV can self-moderate. Given the fact that mandated hospital payment grace periods also set a sufficient t before final payment, such a liquidity pool offsets losses till a critical mass of volume is aggregated. As a result, a revolving line of credit, mapped proportionally to some % of predicted GMV (in most cases, < 20% of predicted GMV), provides a satisfactory solution to such liquidity constraints so long as icredit < iαC.

Interest Rate Variance

In this subsection, we explore some of the possibilities that a potential system could use to generate interest. We do not include a detailed risk analysis in this section, as the management of capital float is independent of price-setting; we have solved our equation open-form, with a variable i. However, we believe it is relevant to recognize the breadth of the option-set here, as circa the publication of this paper, novel yield generation strategies (which may risk-adjust more efficiently) have emerged.

1. Yield Farming: This investment strategy leverages the explosive growth of decentralized finance (DeFi), involving the lending or staking of cryptocurrency to receive rewards in the form of transaction fees and interest. Proof of stake has revolutionized the future of “stable investing returns”, with current yield farming strategies outperforming the Web 2.0 capital markets frequently—often with less volatility (though centrally uninsured). There are implementation complexities that arise from on-ramping and off-ramping USD from any DeFi capital float accounts, but promisingly, many of these have been tested and mitigated at scale by large institutions. We list some of these yield farming strategies, references attached, below:
   • Anchor Protocol Staking
   • Curve Finance StableSwap
   • “BlockFi Yield”
2. Money Markets: Of course, classic capital management strategies work effectively as well. These strategies are “tried and true” in the capital markets, and we list them below:
   • Active: Corporate Portfolio Management
   • Passive: Index Funds & ETFs
   • Passive: Third-Party, Institutional Capital Management

Some hybrid of these yield generation sets may prove most optimal, as long as we satisfy the constraint that icredit < iαC across the same time period. It is relevant to note here that any tail-risk related to interest rate variance is naturally controlled, unlike other industries with capital float: pricing accuracy is tethered to public data points, the federal government has set the ceiling for care charges, and all prices are dynamically set, iterating quickly over time. We explore the last concept in the following subsection.

Given the reliance of our open-form model on a historical heuristic, to best improve system accuracy over time, it is fundamental to adapt our set price z in real-time per (a) the ingestion of new claims data and (b) interest rate variance. This decreases likelihood of confounding from historical-current variance and the tail-risk from interest shocks (from market crashes) to the liquidity pool and GMV—minimizing Value at Risk (VaR) over time.

To accomplish this, we propose the implementation of a simple back-propagation technique, adjusting each claim weight to the network in proportion to how much it contributes to error in the weighted-average price. We can rely on memoization (saved re-computation) to efficiently capture these nested changes over reasonable claims volumes. As a result, our algorithmic balancing becomes more efficient at accurately determining z and reducing incurred risk (precipitously) over time. As below, where xn denotes real-time final bills, Y is our new f(x), and the hidden layer derives our prior f(x) from historical claims nodes: